This paper introduces the unsupervised assignment flow that couples the assignment flow for supervised image labeling (Åström et al. in J Math Imaging Vis 58(2):211–238, 2017) with Riemannian gradient flows for label evolution on feature manifolds. The latter component of the approach encompasses extensions of state-of-the-art clustering approaches to manifold-valued data. Coupling label evolution with the spatially regularized assignment flow induces a sparsifying effect that enables to learn compact label dictionaries in an unsupervised manner. Our approach alleviates the requirement for supervised labeling to have proper labels at hand, because an initial set of labels can evolve and adapt to better values while being assigned to given data. The separation between feature and assignment manifolds enables the flexible application which is demonstrated for three scenarios with manifold-valued features. Experiments demonstrate a beneficial effect in both directions: adaptivity of labels improves image labeling, and steering label evolution by spatially regularized assignments leads to proper labels, because the assignment flow for supervised labeling is exactly used without any approximation for label learning.

中文翻译：

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无监督分配流程：通过空间正则化几何分配在特征流形上进行标签学习

本文介绍了无监督的分配流程结合了监督图像标记的分配流程（Åström等人，J Math Imaging Vis 58（2）：211–238，2017）和黎曼梯度流，用于特征流形上的标签演化。该方法的后一部分包括将最新的聚类方法扩展到多值数据。将标签演变与空间规则化分配流耦合在一起会产生稀疏效果，从而可以以无人监督的方式学习紧凑的标签词典。我们的方法减轻了监督性标签在手边具有适当标签的需求，因为在分配给给定数据的同时，初始标签集可以演变并适应更好的值。特征歧管和分配歧管之间的分离实现了灵活的应用，这在具有歧管值特征的三种情况下得到了证明。实验证明了在两个方向上的有益效果：标签的适应性改善了图像标签，并且通过空间正则化的分配控制标签的演变导致了正确的标签，因为监督标签的分配流程被精确地使用，而标签学习没有任何近似值。